# Sergei Yakovenko's blog: on Math and Teaching

## Dynamics generated by finitely generated subgroups of conformal germs

1. Generic subgroups of $\text{Diff}(\mathbb C^1,0)$ are non-solvable.
2. Dynamics generated by several germs. Definition of a pseudogroup. Orbits of points.
3. Periodicity of germs (finiteness of order) vs. periodicity of orbits. Cycles and limit cycles of pseudogroups.
4. Convergence of elements in pseudogroups. Closure.
5. Density of orbits. Linear subgroups. Abundance of limit cycles for generic (nonsolvable) subgroups of $\text{Diff}(\mathbb C^1,0)$.
6. Topological equivalence of subgroups and pseudogroups. Conjugacy of dense linear subgroups.
7. Rigidity of nonsolvable subgroups: topological conjugacy implies holomorphic conjugacy.

Disclaimer… if somebody still needs it… 😦
Reading: Section 6 (second part) from the book, printing disabled.

1. In the definition of a cycle (Def. 6.31) nontriviality of an element probably needs a clarification: in a (pseudo)group $G$ generated by non-identical germs $f_1,\dots,f_n\in\text{Diff}(\mathbb C^1,0)$ an element is nontrivial if it corresponds to a nontrivial word $w$ in the free group in $n$ symbols. Thus the identical germ $\text{Id}$ is nontrivial if and only if $G$ admits nontrivial identities.